212 
CALCULUS 
For convenience, let a = + oo, the interval then becoming 
g^x<cc. By the Generalized Law of the Mean, § 1, we have 
fix)-Ax’) f'(X) 
g < x' < x, x' < X < x. 
F.(x)-F(x') F'(X) 
Hence 
X _l-F(x')/F(x) 
1 ~f( x ')/f( x ) 
Now let x and x' both become infinite; but let x increase so much 
more rapidly that 
Then lim X — 1. How, X becomes infinite, and hence the whole 
right-hand side of the first equation (3) approaches the limit which 
f'(x)JF'(x) by hypothesis approaches. Thus the theorem is proved.* 
If f'(x)/F\x) becomes positively infinite, or negatively infinite, 
the same is true of f(x)/F(x). 
If a is a finite point, the same reasoning still holds, with obvious 
modifications in details. Or, this case can be referred directly to 
the above by means of such a substitution as 
V = VO - a ), 
x = a + 1/y. 
This theorem has the same advantage as that of § 1, namely, that, 
if we do not get a result after the first pair of differentiations, we 
may differentiate again and again. If, after k repetitions, we do 
get a result, then the original ratio approaches this same limit. 
• QC™ 
Example 1. lim 
X=oo S X 
If n fg 0, the limit is obviously 0, since the numerator remains 
finite and the denominator becomes infinite. If, however, n > 0, we 
see that the above ratio approaches 0 provided n ^ 1. If n > 1, a 
finite number of repetitions will lead to a ratio whose limit is 0, and 
thus the given ratio approaches 0 for any fixed value of n. 
Example 2. lim - , 0 < a, 0 < ¡8. 
If a — 1, we have 
lim = li m JL — o. 
x=oo X^ ac=co j3x^ 
* The theorem is due to Cauchy, who gave a proof under certain restrictions. 
The complete prt>of, given above, is due to the Austrian mathematician Stolz. '